login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A133953 A second integer solution:d=2;h=1; A 4 X 4 vector Markov of a game matrix MA and an anti- game matrix MB such that game_valueMa+game_ValueMB =0 and the score is the sum of the vector out put of the Markov: MA={{0,1},{1,d}}; MB={{1/h,0},(2 - d + 1/h + h),h}}; Characteristic Polynomial is: -1 + 4 x^2 - 4 x^3 + x^4. 0
2, 6, 12, 24, 50, 110, 252, 592, 1410, 3382, 8140, 19624, 47346, 114270, 275836, 665888, 1607554, 3880934, 9369356, 22619576, 54608434, 131836366, 318281084, 768398448, 1855077890, 4478554134, 10812186060, 26102926152, 63018038258 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Using the general game value function: gv[M_] = Det[M]/Sum[Sum[M[[i, j]]*(-1)^(i + j), {i, 1, 2}], {j, 1, 2}] and matrices: MA = {{a, b}, {c, d}}; MB = {{e, f}, {g, h}}; The solution of: Solve[{b == 1, f == 0, a == 0, Det[MA] + 1 == 0, Det[MB] - 1 == 0, gv[MA] + gv[MB] == 0}, {c, d, e, g, h}] Gives the two Matrix solution: MA={{0,1},{1,d}}; MB={{1/h,0},{2-d+1/h+h,h}}; Besides the Fibonacci:d=1 and h=1 the only other obvious integer solution is: d=2 and h=1.

LINKS

Table of n, a(n) for n=1..29.

FORMULA

d = 2; h = 1; M = {{0, 1, 0, 0}, {1,d, 0, 0}, {0, 0, 1/h, 0}, {0, 0, 2 - d + 1/h + h, h}}; v[1] = {0, 1, 1, 0}; v[n_] := v[n] = M.v[n - 1] a(n) = Sum[v(i),{i,1,4}].

G.f.: 2*x*(1+x)*(1-2*x)/((1-x)^2*(1-2*x-x^2)). [Colin Barker, Feb 28 2012]

MATHEMATICA

d = 2; h = 1; M = {{0, 1, 0, 0}, {1, d, 0, 0}, {0, 0, 1/h, 0}, {0, 0, 2 - d + 1/h + h, h}}; v[1] = {0, 1, 1, 0}; v[n_] := v[n] = M.v[n - 1] a = Table[Apply[Plus, v[n]], {n, 1, 50}]

CROSSREFS

Sequence in context: A309841 A132176 A197469 * A122863 A170935 A277173

Adjacent sequences:  A133950 A133951 A133952 * A133954 A133955 A133956

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Jan 08 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 07:03 EST 2019. Contains 329978 sequences. (Running on oeis4.)